Yes, you can quit claim him a joint ownership interest, but be sure doing so will not cause a default under your loan. The only way to be sure of this is to carefully review all of the mortgage documentation (e.g., mortgage, promissory note, etc.). You should also be mindful of any potential tax consequences. It's best to work with a lawyer on this task.

It is actually pretty common at real estate closings for one spouse to buy the house in their name because they would not qualify for the mortgage if the other spouse was on the loan. But, the intent was always that they co-own the house. The paperwork to create the joint ownership is routinely prepared by the title company and executed as part of the closing. As long as you are creating the joint ownership with an immediate relative (like your son) and you retain partial ownership, you should not have a problem. But, as the previous answer said, you should carefully review your loan documents and make sure you file all of the right paperwork with the county relating to the transfer. If you are not certain that you can do this yourself, have an attorney do it to make sure you do not create any problems for yourself.